Ch 5 0910

Roles of Wiki	Student #1	Student #2
Editor	Leann Murphy
Multimedia	Evan Newcomer
Text	Alisha Quigley	Rachel Robinson
Practice Problems	Joe Patterson
Career/Real Life App	Christian Molander

Section 1: Graphing Quadratic Equations

Vocabulary

• Quadratic Function: y = ax² + bx + c
• Parabola: graph of a quadratic function; U-shaped
• Vertex: lowest or highest point on the parabola
• Axis of Symmetry: Vertical line that passes through the vertex;

x = - b / 2a

Forms of Graphing & How to Graph them

- Standard Form

1. y = ax² + bx + c
2. Direction of graph: upward if a > 0, downward if a < 0
3. Axis of Symmetry: x = - b / 2a
4. Vertex: (x,y) X is found using, - b / 2a . Y is found by plugging the x-value back into the equation.
5. Plot two points plugging x-values into the equation and using the axis of symmetry to plot them on the other side.

- Vertex Form

1. y = a(x - h)² + k
2. Direction of graph: upward if a > 0, downward if a < 0
3. Axis of Symmetry: x = k
4. Vertex: (h,k)
5. Plot two points plugging x-values into the equation and using the axis of symmetry to plot them on the other side.

- Intercept Form

1. y = a(x - p)(x - q)
2. Direction of graph: upward if a > 0, downward if a < 0
3. Axis of Symmetry: halfway between (p,0) and (q,0)
4. Vertex: ( [axis of symmetry], y) Y is found plugging the x-value back into the equation
5. X-intercepts: p and q

To turn Vertex or Intercept Form into Standard Form

• F O I L
  • multiply the First term, Outer term, Inner term, and Last term, then combine like terms.

Real Life Application
You can model real-life archeing objects with Parabolas, such as the golden gate bridge.

Practice:
Graph y = 2x^2 – 8x + 6
Label Vertex axis of symmetry and two other points.

Section 2: Solving Quadratic Equations by Factoring

Vocabulary

• Monomial: expression that has one term
• Binomials: expression that has two terms
• Trinomial: expression that has three terms
• Factoring: processed used to separate trinomials as a product of binomials
• Zeros: The numbers p and q are called zeros in the equation y=a(x-p)(x-q) ; x = p and x = q

Factoring Trinomials
To factor x² + bx + c

1. Find integers m and n
2. x² + bx + c = (x + m)(x + n) = (x² + mx) + (nx + mn)
3. The sum of m and n must equal b, and the product of m and n must equal c

To factor ax ² + bx + c

1. Find integers k and l, as well as m and n
2. ax² + bx + c = (kx + m)(lx+n) = (klx² + kn) + (lmx + mn)
3. k and l must be factors of a, and m and n must be factors of c

Factoring Patterns

• Difference of Two Squares: a² - b² = (a +b)(a - b)
• Perfect Square Trinomial: a² + 2ab + b² = (a + b)²

Zero Product Property

• ax² + bx + c = 0

Finding Zeros

1. Change equation into Intercept Form
2. Factor as usual
3. When you reach, (x ± # )(x ± #) , replace x-value in each group with the value opposite of the # so that when added together they should equal zero
4. The x-value in each grouping is the zero

Real Life Application
You can use quadratic equations in different jobs, such as finding appropriate dimensions for things like paintings, landmarks, etc.

Practice:
Factor 3x^2 - 17x + 10

Section 3: Solving Quadratic Equations by Finding Square Roots

Vocabulary

• Square Root: a number r is a square root of a number s; r² = s , √s = r
• Radical Sign: √
• Radicand: the number or expression beneath the radical sign
• Radical: the expression √s where s is a number
• Rationalizing the Denominator: process of eliminating a radical in the denominator of a fraction by multipling both the denominator and the numerator by denominator

Properties of Square Roots

1. Product Property:√ab = (√a)(√b)
2. Quotient Property: √a/b = √a/b

Solving Quadratic Equations using Square Roots

1. Write original equation
2. Get to the point where only the x²-values are on one side and only a number is on the other side
3. Take the square root of both sides
4. Simplify
5. Remember the answer will be both ±

Finding a Falling Object's Height using Quadratic Function

1. h = -16t² + h₀
2. h is the feet; t is the seconds after the objects dropped; h₀ is the initial height

Real Life Application
You can model real life quantities, such as the height of a rock dropped off the Leaning Tower of Pisa

Practice:

Slove 2x^2 + 1 = 17

Section 4: Complex Numbers

Vocabulary

• Imaginary Unit: defined as i. i = √-1 , i² = -1
• Complex Number: a + bi where a and b are real numbers and i is the imaginary unit; a is always the real part of the complex number and bi is the imaginary part
• Imaginary Number: complex number a + bi where b ≄ 0
• Complex Plane: Coordinate plane where each point is (a,b) represents a + bi. Horizontal axis is for real numbers and vertical axis is for imaginary numbers.
• Complex Conjugates: Two complex numbers, a + bi and a - bi. The product is always a real number

Square Root of Negative Numbers

1. If r is a positive real number, then √-r = i√r
2. If r is a postive real number and follows the above, then (i√r)² = -r

Solving a Quadratic Equation

1. Write original equation
2. Get to the point where only the x²-values are on one side and only a number is on the other side
3. Take the square root of each side
4. Write in terms of i when taking the square root of a negative number
5. Simplify the radical
6. Remember the answer will be both ±

Plotting Complex Numbers

1. The normal x-axis is now the Real Number axis, like wise the normal y-axis is now the Imaginary Number axis
2. Starting at the origin; the a value of the equation will follow the Real Number axis, the b value of the equation will follow the Imaginary Number Axis.

Multiplying Complex Numbers

1. Use the Distributive Property or FOIL
2. Get all the number and values on to one side
3. Simplify and use i² = -1
4. Answer should be in standard form; a ± bi

Dividing Complex Numbers

1. Multiply the denominator and the numerator by the conjugate
2. Use FOIL
3. Simplify
4. Answer should be in standard form; a ± bi

Absolute Values of Complex Numbers

• |z| = √a² + √b²
• Used to find the number's distance from the origin in the complex plane
• When solving the i-value is disregarded

Real Life Application
You can use complex numbers to solve problems, such as determining whether a complex number belongs to the Mandelbrot set

Practice:

Slove 3x^2 + 10 = -26

Section 5- Completing The Square

Completing The Square: process that allows you to write an expression of the form x² + bx as the square binomial
c = (b/2)²

Solving quadratic equation when coefficient of x² is 1:
1) set equation to x² + bx = c
2) add (b/2)² to both sides
3) factor into (x + b/2)²
4) square root both sides.
5) solve for x

Solving quadratics when coefficient of x² is not 1:
1) divide each side by coefficient of x²
2) set equation to x² + bx = c
3) add (b/2)² to both sides
4) factor into (x + b/2)²
5) square root both sides
6) solve for x


Real Life Application-
This is good for firefighters, because it can help them figure out where to position a fire hose where it would be most effective against the fire

Practice:
Slove x^2 + 10x - 3 = 0 by completing the square


Section 6- Quadratic Formula and Discriminant

Quadratic Formula- x = -b ± √b² - 4ac / 2a
Discriminant- b² - 4ac
If:
b² - 4ac > 0 ; 2 real solutions
b² - 4ac < 0 ; 2 imaginary solutions
b² - 4ac = 0 ; 1 real solution





Real Life Application-
This is good if you own an amusement park, because you can find the speed and duration of thrill rides by using the quadratic formula and the discriminant

Practice:

Solve 2x + x = 5


Section 7- Graphing and Solving Quadratic Inequalities

Quadratic inequalities with 2 variables: y>ax² + bx + c ; y>ax² + bx + c ; y<ax² + bx + c ; y<ax² + bx + c
Quadratic inequalities with 1 variable: ax² + bx + c<0 ; ax² + bx + c<0 ; ax² + bx + c> 0 ; ax² + bx + c> 0

1) Graph as you would a regular equation, remembering to use dashed or solid lines
2) pick a point and plug it in the equation to see if it's true or false.
3) shade on the appropriate side

Shade the region that makes the equation true.
When shading:
If shading the outside- OR statement (ex. x< -2 or x> 1)
If shading inside- AND statement (ex. -2 < x <1)

Real Life Application-
You can solve real life problems like finding the weight of theater equipment that certain kinds of rope can support without breaking

Practice:

Graph y > x^2 - 2x - 3
Label Vertex, Axis and two other points.

Section 8- Modeling with Quadratic Functions
Quadratic: function that represents a real data set
Vertex Form: y = a(x-h)² + k
Intercept Form: y = a(x-p)(x-q)

Taking the vertex & given point and putting it into an equation:
1) Take the vertex points and put it in the equation so it's y = a(x-h)² + k
2) Substitute the given point for x and y
3) Solve for the value of a.
4) Write the equation

Using intercepts to write an equation:
1) Put the intercepts into the equation
2) Substitute the given point for x and y
3) solve for a
4) write equation

Real Life Application-
You can solve important problems like determining the effect of wind on a runner's performance

Practice:

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Answers:
Vertex: (2,-2) Axis: x=2
(3x – 2)(x – 5)
2 square roots of 2 and -2 square roots of 2
X = + or 2i square roots of 3
X = -5 + or – 2 square roots of 7
X = about 1.35 and x = about -1.85
Vertex: (1,4) Axis: x = 1 Other Points: (-1,0) and (3,0)